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  "body": "\n## Summary\n\nA divisibility rule is a short method to decide whether a fixed integer $d$ divides an integer $n$.\nIt uses the digits of $n$, and it does not need a full division.\nMost rules for base $10$ come from the [[congruences|0TPW25CBN6]] of the powers $10^k$ modulo $d$.\nFor example, $10 \\equiv 1 \\pmod 3$, so $n$ and the sum of its digits have the same remainder modulo $3$.\n\n## Remarks\n\nFor example, $2$ divides $n$ if and only if the last digit of $n$ is even. Also $5$ divides $n$ if and only if the last digit is $0$ or $5$.\nSee [[divisibility by 3|0R9D4CEJEB]], [[divisibility by 9|0ESHD7ZQDX]] and [[divisibility by 11|09YBMJTNBM]].\n[[Casting out nines|0E7W0EM6J6]] uses the rule for $9$ to check computations.\n\n## Explanations\n\n### L1 human\n\nYou can see at once that $1{,}234{,}570$ is a multiple of $10$, because it ends with $0$.\nDivisibility rules give similar quick tests for other divisors, for example $3$, $4$, $9$ and $11$.\n",
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